If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Class 12 math (India)

### Course: Class 12 math (India)>Unit 10

Lesson 11: Riemann sums

# Riemann sums worksheet

Practice identifying and calculating Riemann sums.

## Part 1: The three types of Riemann sums

1) What type of Riemann sum is shown below?

2) What type of Riemann sum is shown below?

3) What type of Riemann sum is shown below?

## Part 2: Left Riemann sum

The diagram below shows the left Riemann sum. We want to find the total area of the four rectangles.
The first rectangle: The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 0, right parenthesis, equals, 1, plus, 0, point, 1, dot, 0, squared, equals, start color #1fab54, 1, end color #1fab54 unit. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 1, end color #1fab54, equals, 2 unitssquared.
The second rectangle: The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 2, right parenthesis, equals, 1, plus, 0, point, 1, dot, 2, squared, equals, start color #1fab54, 1, point, 4, end color #1fab54 units. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 1, point, 4, end color #1fab54, equals, 2, point, 8 unitssquared.
The third rectangle: The base is start color #11accd, 2, end color #11accd units. The height is f, left parenthesis, 4, right parenthesis, equals, 1, plus, 0, point, 1, dot, 4, squared, equals, start color #1fab54, 2, point, 6, end color #1fab54 units. The area is start color #11accd, 2, end color #11accd, dot, start color #1fab54, 2, point, 6, end color #1fab54, equals, 5, point, 2 unitssquared.
The fourth rectangle:
The base is
units.

The height is
units.

The area is
unitssquared.

Now add up the areas of the four rectangles to get the left Riemann sum approximation for the area under the graph of the function f, left parenthesis, x, right parenthesis, equals, 1, plus, 0, point, 1, x, squared on the interval open bracket, 0, comma, 8, close bracket.
unitssquared

## Part 3: Midpoint Riemann sum

The diagram below shows the midpoint Riemann sum.
Which is an expression for the midpoint Riemann sum?

## Part 4: Right Riemann sum

The diagram below shows the right Riemann sum.
Which is an expression for the right Riemann sum?

## Want to join the conversation?

• Left Riemann sum is f(left)+f(left+base) and so on?
• Yes! It appears to be. (I'm assuming you know to multiply the result by delta-x or base)

It would be helpful to have a formal definition of them attached to this worksheet or introduced beforehand. I haven't seen them before this. (I have been following the Integral Calculus course outline.
• Midpoint reimann sum gives a better approximation than left-hand and right-hand.So,what is the point of using left-hand and right-hand reimann sums?
• They are used to estimate an integral (area under the curve)
• i wnat defination of bonded function ,partition of I=[a,b],upper RIEMANN AND LOWER RIEMANNof partition
• it appears to me that the mid point Riemann sum gives the closest aproxximation
is this observation correct in general??